A spectral boundary integral method for simulating electrohydrodynamic flows in liquid droplets*

A weakly conducting liquid droplet immersed in another leaky dielectric liquid can exhibit rich dynamical behaviors under the effect of an applied electric field. Depending on material properties and field strength, the nonlinear coupling of interfacial charge transport and fluid flow can give rise to various electrohydrodynamic instabilities leading to shape deformations and complex dynamics. Here, we present a spectral boundary integral method for the simulation of droplet electrohydrodynamics in uniform applied fields. All physical variables, such as drop shape and interfacial charge density, are represented using spherical harmonic expansions. In addition to its excellent accuracy, the spectral representation affords a nondissipative de-aliasing method required for numerical stability. We also use a reparametrization technique, which we find required to explore regimes where the drop undergoes significant deformations. Our simulations closely match existing experimental data and past analytical predictions in both the axisymmetric Taylor and Quincke electrorotation regimes. Moreover, we demonstrate that the dynamics of low-viscosity drops are strongly affected by charge convection by the flow, which results in the emergence of steep gradients in the interfacial charge density.